> > If you believe that the real unit interval together with + and * is> > not isomorphic to the real unit interval with + and * then you may do> > so . I call them isomorphic.>> What I said was that the real with interval with + is not a group.>> > If ax + by is in the unit interval, then f(ax + by) is in the tree.>> But ax + by is NOT always in the unit interval, so accordingly f(ax +> by) need not be in the tree.>> > If ax + by is not in the unit interval, then f(ax + by) is not in the> > tree.>> But for a mapping to be linear on the unit interval requires that for> any x and y in the unit interval and any a and b in the field of scalars> ax+by also be in the interval. Otherwise the interval is not a linear> space at all and there cannot be any linear mappings from it to anything.